| name | upper case | lower case | usage |
|---|---|---|---|
| Lambda | \(\Lambda\) | \(\lambda\) | Loading of a manifest indicator onto a latent construct |
| Psi | \(\Psi\) | \(\psi\) | residual variance/covariance of contruct when endogenous |
| Theta | \(\Theta\) | \(\theta\) | residual variance/covariance of indicators |
| Sigma | \(\Sigma\) | \(\sigma\) | \(\Sigma\) is the model implied variance/covariance matrix; |
| \(\sigma\) is standard deviation, \(\sigma^2\) variance of indicator. | |||
| \(\sigma\) can also be covariance of indicator |
\[ \textbf{$\Lambda$} = \left[ \begin{array}{cc} \lambda_{1,1} \\ \lambda_{2,1} \\ \lambda_{3,1} \end{array} \right],\]
\[ \textbf{$\Psi$} = \left[ \begin{array}{cc} \psi_{1,1} \end{array} \right],\]
\[ \textbf{$\Lambda^\prime$} = \left[ \begin{array}{cc} \lambda_{1,1} & \lambda_{2,1} & \lambda_{3,1} \end{array} \right],\]
\[ \textbf{$\Theta$} = \left[ \begin{array}{cccccc} \theta_{1,1} & 0 & 0 \\ 0 & \theta_{2,2} & 0 \\ 0 & 0 & \theta_{3,3} \end{array} \right].\]
\[ \Sigma = \Lambda \Psi \Lambda' + \Theta \tag{1} \]
library(lavaan)
##Prepare data with sufficient statisitics##
mymeans<-matrix(c(3.06893, 2.92590, 3.11013), ncol=3,nrow=1)
mysd<-c(0.84194,0.88934,0.83470)
mat <- c(1.00000,
0.55226, 1.00000,
0.56256, 0.60307, 1.00000)
mycor <- getCov(mat, lower = TRUE)
##Transform correlation matrix to covariance matrix using information above##
mycov <- mysd %*% t(mysd)
rownames(mycor) <-c( "Glad", "Cheerful", "Happy")
colnames(mycor) <-c( "Glad", "Cheerful", "Happy")
rownames(mycov) <-c( "Glad", "Cheerful", "Happy")
colnames(mycov) <-c( "Glad", "Cheerful", "Happy")
mynob<-823
| Glad | Cheerful | Happy | |
|---|---|---|---|
| Glad | 1.00 | 0.55 | 0.56 |
[1] MODEL RESULTS
[2] Two-Tailed
[3] Estimate S.E. Est./S.E. P-Value
[4] POSITIVE BY
[5] GLAD1 1.000 0.000 999.000 999.000
[6] CHEER1 1.072 0.061 17.612 0.000
[7] HAPPY1 1.092 0.062 17.624 0.000
[8] Variances
[9] POSITIVE 0.515 0.049 10.521 0.000
[10] Residual Variances
[11] GLAD1 0.484 0.033 14.547 0.000
[12] CHEER1 0.408 0.033 12.211 0.000
[13] HAPPY1 0.385 0.034 11.484 0.000
lambda = matrix(c(1.00, 1.072, 1.092), nrow = 3)
lambda
[,1]
[1,] 1.000
[2,] 1.072
[3,] 1.092
psi = matrix(.515)
psi
[,1]
[1,] 0.515
t(lambda)
[,1] [,2] [,3]
[1,] 1 1.072 1.092
theta = diag(c(.484, .408, .385))
theta
[,1] [,2] [,3]
[1,] 0.484 0.000 0.000
[2,] 0.000 0.408 0.000
[3,] 0.000 0.000 0.385
sigma = lambda %*% psi %*% t(lambda) + theta
round(sigma,2)
[,1] [,2] [,3]
[1,] 1.00 0.55 0.56
[2,] 0.55 1.00 0.60
[3,] 0.56 0.60 1.00
| Glad | Cheerful | Happy | |
|---|---|---|---|
| Glad | 1.00 | 0.55 | 0.56 |
\[ \textbf{$\Sigma$} = \left[ \begin{array}{cccccc} \sigma_{1,1}^2 & \sigma_{1,2} & \sigma_{1,3} & \sigma_{1,4} & \sigma_{1,5} & \sigma_{1,6} \\ \sigma_{2,1} & \sigma_{2,2}^2 & \sigma_{2,3} & \sigma_{2,4} & \sigma_{2,5} & \sigma_{2,6} \\ \sigma_{3,1} & \sigma_{3,2} & \sigma_{3,3}^2 & \sigma_{3,4} & \sigma_{3,5} & \sigma_{3,6} \\ \sigma_{4,1} & \sigma_{4,2} & \sigma_{4,3} & \sigma_{4,4}^2 & \sigma_{4,5} & \sigma_{4,6} \\ \sigma_{5,1} & \sigma_{5,2} & \sigma_{5,3} & \sigma_{5,4} & \sigma_{5,5}^2 & \sigma_{5,6} \\ \sigma_{6,1} & \sigma_{6,2} & \sigma_{6,3} & \sigma_{6,4} & \sigma_{6,5} & \sigma_{6,6}^2 \end{array} \right],\]
\[ \textbf{$\Lambda$} = \left[ \begin{array}{cc} \lambda_{1,1} & 0 \\ \lambda_{2,1} & 0 \\ \lambda_{3,1} & 0 \\ 0 & \lambda_{4,2} \\ 0 & \lambda_{5,2} \\ 0 & \lambda_{6,2} \end{array} \right],\]
\[ \textbf{$\Psi$} = \left[ \begin{array}{cc} \psi_{1,1} & \psi_{1,2} \\ \psi_{2,1} & \psi_{2,2} \end{array} \right],\]
\[ \textbf{$\Lambda^\prime$} = \left[ \begin{array}{cc} \lambda_{1,1} & \lambda_{2,1} & \lambda_{3,1} 0 & 0 & 0\\ 0 & 0 & 0 & \lambda_{4,2} & \lambda_{5,2} & \lambda_{6,2} \end{array} \right],\]
\[ \textbf{$\Theta$} = \left[ \begin{array}{cccccc} \theta_{1,1} & 0 & 0 & 0 & 0 & 0 \\ 0 & \theta_{2,2} & 0 & 0 & 0 & 0 \\ 0 & 0 & \theta_{3,3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \theta_{4,4} & 0 & 0 \\ 0 & 0 & 0 & 0 & \theta_{5,5} & 0 \\ 0 & 0 & 0 & 0 & 0 & \theta_{6,6} \end{array} \right].\]
lambda = matrix(c(.712, .788, .768, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, .729, .764, .778), nrow = 6)
lambda
[,1] [,2]
[1,] 0.712 0.000
[2,] 0.788 0.000
[3,] 0.768 0.000
[4,] 0.000 0.729
[5,] 0.000 0.764
[6,] 0.000 0.778
psi = matrix(c(1.00, 0.561,
0.561, 1.00), nrow = 2)
psi
[,1] [,2]
[1,] 1.000 0.561
[2,] 0.561 1.000
theta = diag(c(.491, .378, .409, .467, .416, .394))
theta
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.491 0.000 0.000 0.000 0.000 0.000
[2,] 0.000 0.378 0.000 0.000 0.000 0.000
[3,] 0.000 0.000 0.409 0.000 0.000 0.000
[4,] 0.000 0.000 0.000 0.467 0.000 0.000
[5,] 0.000 0.000 0.000 0.000 0.416 0.000
[6,] 0.000 0.000 0.000 0.000 0.000 0.394
sigma = lambda %*% psi %*% t(lambda) + theta
round(sigma, 2)
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 1.00 0.56 0.55 0.29 0.31 0.31
[2,] 0.56 1.00 0.61 0.32 0.34 0.34
[3,] 0.55 0.61 1.00 0.31 0.33 0.34
[4,] 0.29 0.32 0.31 1.00 0.56 0.57
[5,] 0.31 0.34 0.33 0.56 1.00 0.59
[6,] 0.31 0.34 0.34 0.57 0.59 1.00